Difference between revisions of "Group action"
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| − | {{ | + | {{Stub page|grade=A|msg=Needs fleshing out and neatening up, I'd like to introduce right group actions in a different way to left, however in my current attempt they're the same length!}} |
| − | == | + | __TOC__ |
| − | A ''group action'' of a group {{M|G}} on a set {{M| | + | ==Defintion== |
| + | A (left) ''group action'' of a [[group]] {{M|(G,*)}} on a [[set]] {{M|X}} is a [[mapping]]{{rAAPAG}}: | ||
| + | * {{M|(\cdot):G\times X\rightarrow X}}<ref group="Note">I have written {{M|(\cdot):G\times X\rightarrow X}} rather than the usual {{M|\cdot:G\times X\rightarrow X}} notation for [[function|functions]] to make it clearer that there is a dot there; this notation isn't new or different, it's just because a lone {{M|\cdot}} looks out of place.</ref> defined by {{M|(\cdot):(g,x)\mapsto g\cdot x}} such that: | ||
| + | ** {{M|1=\forall x\in X[1\cdot x=x]}} (where {{M|1}} is the identity element of {{M|(G,*)}} group) and | ||
| + | ** {{M|1=\forall g,h\in G\ \forall x\in X[g\cdot(h\cdot x)=(g*h)\cdot x]}} | ||
| + | Notations for {{M|g\cdot x}} include {{M|gx}} and {{M|{}^gx}} | ||
| − | |||
| + | A ''right group action''<ref name="AAPAG"/> is almost exactly the same, just the other way around; defined by {{M|(\cdot):X\times G\rightarrow X}} given by {{M|(\cdot):(x,g)\mapsto x\cdot g}} which must satisfy {{M|1=\forall x\in X[x\cdot 1=x]}} and {{M|1=\forall g,h\in G\ \forall x\in X[(x\cdot g)\cdot h=x\cdot(g*h)]}}. | ||
| + | |||
| + | Notations for {{M|x\cdot g}} include {{M|xg}} and {{M|x^g}} | ||
| + | ==See also== | ||
| + | * '''Examples:''' | ||
| + | ** [[Every group acts on itself by multiplication]] | ||
| + | ** [[Every subgroup of a group acts on the group by multiplication]] | ||
| + | ** [[The symmetric group on a set acts on the set by evaluation]] | ||
| + | * [[Any group action can be thought of as a permutation]] | ||
| + | ==Notes== | ||
| + | <references group="Note"/> | ||
| + | ==References== | ||
| + | <references/> | ||
| + | {{Group theory navbox|plain}} | ||
| + | {{Abstract algebra navbox}} | ||
{{Definition|Abstract Algebra|Group Theory}} | {{Definition|Abstract Algebra|Group Theory}} | ||
Latest revision as of 23:28, 21 July 2016
Stub grade: A
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Needs fleshing out and neatening up, I'd like to introduce right group actions in a different way to left, however in my current attempt they're the same length!
Contents
Defintion
A (left) group action of a group [ilmath](G,*)[/ilmath] on a set [ilmath]X[/ilmath] is a mapping[1]:
- [ilmath](\cdot):G\times X\rightarrow X[/ilmath][Note 1] defined by [ilmath](\cdot):(g,x)\mapsto g\cdot x[/ilmath] such that:
- [ilmath]\forall x\in X[1\cdot x=x][/ilmath] (where [ilmath]1[/ilmath] is the identity element of [ilmath](G,*)[/ilmath] group) and
- [ilmath]\forall g,h\in G\ \forall x\in X[g\cdot(h\cdot x)=(g*h)\cdot x][/ilmath]
Notations for [ilmath]g\cdot x[/ilmath] include [ilmath]gx[/ilmath] and [ilmath]{}^gx[/ilmath]
A right group action[1] is almost exactly the same, just the other way around; defined by [ilmath](\cdot):X\times G\rightarrow X[/ilmath] given by [ilmath](\cdot):(x,g)\mapsto x\cdot g[/ilmath] which must satisfy [ilmath]\forall x\in X[x\cdot 1=x][/ilmath] and [ilmath]\forall g,h\in G\ \forall x\in X[(x\cdot g)\cdot h=x\cdot(g*h)][/ilmath].
Notations for [ilmath]x\cdot g[/ilmath] include [ilmath]xg[/ilmath] and [ilmath]x^g[/ilmath]
See also
Notes
- ↑ I have written [ilmath](\cdot):G\times X\rightarrow X[/ilmath] rather than the usual [ilmath]\cdot:G\times X\rightarrow X[/ilmath] notation for functions to make it clearer that there is a dot there; this notation isn't new or different, it's just because a lone [ilmath]\cdot[/ilmath] looks out of place.
References
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